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Gromov-Witten invariants on Grassmannians. (English) Zbl 1063.53090
The main result of the paper under review is: any three-point genus zero Gromov-Witten invariant on a Grassmannian $$X$$ is equal to a classical intersection number on a homogeneous space $$Y$$ of the same Lie type. The statement was proven when $$X$$ is a type $$A$$ Grassmannian, and, in types $$B$$, $$C$$, and $$D$$, when $$X$$ is the Lagrangian or orthogonal Grassmannian parametrizing maximal isotropic subspaces in a complex vector space equipped with a non-degenerate skew-symmetric or symmetric form. Their key identity for Gromov-Witten invariants is based on an explicit bijection between the set of rational maps counted by a Gromov-Witten invariant and the set of points in the intersection of three Schubert varieties in the homogeneneous space $$Y$$. It should be noted that the proof of their result does not use moduli spaces of maps and only uses basic algebraic geometry. In types $$B$$, $$C$$, and $$D$$, their result may be used to give new proofs of the structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians obtained by A. Kresch and H. Tamvakis in [J. Algebr. Geom. 12, No. 4, 777–810 (2003; Zbl 1051.53070) and Compos. Math. 140, No. 2, 482–500 (2004; Zbl 1077.14083)]. Their methods can also be used to prove a quantum Pieri rule for the quantum cohomology of sub-maximal isotropic Grassmannians.

##### MSC:
 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14N15 Classical problems, Schubert calculus 05E15 Combinatorial aspects of groups and algebras (MSC2010)
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